By passing to high powers of the \(f_i\) 's we may assume that \(k=k_i\)  for all \(1\le i \le h\) . Fix \(k\)  and put \(I={{\bf a}}^{(k)}\)  and \(J=(f_1,\ldots ,f_{h})\) . We will show that \(J\)  is a reduction of \(I\) . Let \(B=R/(x_1,\ldots ,x_{d-h})\) . Then as \(\underline{x}=(x_1,\ldots ,x_{d-h})\)  is \(R\) -regular, by [1]} and condition (REF ) we have
\(e_J(B)=e_{(\underline{x})+J}(R)= k^{h} \sum _{i=1}^se_{{\bf a}R_{P_i}}(R_{P_i})e_{\underline{x}} \left( Ȓ{P_i}\right) .\) 
Let \(F=\operatorname{MinAss}_R (R/J)\) . Then for all \(n\ge 1\) , \(F=\operatorname{MinAss}_R (R/J^{n})\) . Hence we get
\( \nonumber {e_J(B)}&=& \left[\underset{{ n \rightarrow \infty }}{\lim }{n^{h}} {\lambda }_B(B/J^{n}) \right]\\ \nonumber & = & \underset{{ n \rightarrow \infty }}{\lim }\left[ {n^{h}} {\lambda }_R(R/(\underline{x})+J^{n})\right]\\ \nonumber & \ge & \underset{{ n \rightarrow \infty }}{\lim }{n^{h}} \left[e_{\underline{x}R} \left( Ȓ{J^{n}} \right) \right]\\ \nonumber & = & \underset{{ n \rightarrow \infty }}{\lim }{n^{h}} \left[ \displaystyle {\sum _{P\in F}{\lambda }_{R_P}(R_P/J^{n}R_P)e_{\underline{x}}(R/P)} \right]\mbox{by \cite [Theorem~14.7]{mat}}\\&=&{\displaystyle {\sum _{P\in F}e_{JR_P}(R_P)e_{\underline{x}}(R/P)}}\) 
As \(\operatorname{ht}I=\operatorname{ht}J=h\)  and \(J \subseteq I\)  for all \(i=1, \ldots , s \) , \(P_i\in F\) ,
\(\sum _{P\in F}e_{JR_P}(R_P)e_{\underline{x}}(R/P)\ge \sum _{i=1}^se_{JR_{P_i}}(R_{P_i})e_{\underline{x}}(R/P_i).\) 
Now from () and (REF ) we get
\({e_J(B)}\ge \sum _{i=1}^se_{JR_{P_i}}(R_{P_i})e_{\underline{x}}(R/P_i).\) 
Since \(J\subseteq I\) , for \(1\le i \le s\) ,
\(e_{JR_{P_i}(R_{P_i})} &\ge & e_{IR_{P_i}}(R_{P_i})\) 
Since \(I^{n}R_{P_i} = ({{\bf a}}^{(k)}R_{P_i})^{n}={{\bf a}}^{kn}R_{P_i}\) , we have
\(e_{IR_{P_i}}(R_{P_i})= \underset{{ n \rightarrow \infty }}{\lim }\frac{h!}{n^h}{\lambda }(R_{P_i}/I^{n}R_{P_i})= \underset{{ n \rightarrow \infty }}{\lim }\frac{h!}{n^h}{\lambda }(R_{P_{i}}/{{\bf a}}^{kn}R_{P_i})= k^{h}e_{{{\bf a}}R_{P_i}}(R_{P_i}) .\) 
Thus from (REF ), (REF ), (REF ) and (REF ) we have
\(e_J(B)&\ge &\displaystyle {\sum _{i=1}^se_{JR_{P_i}}(R_{P_i})e_{\underline{x}}(R/P_i)} \\& \ge & \displaystyle {\sum _{i=1}^se_{IR_{P_i}}(R_{P_i})e_{\underline{x}}(R/P_i)}\\& = & \displaystyle {\sum _{i=1}^se_{{{\bf a}}R_{P_i}}(R_{P_i})k^{h}e_{\underline{x}}(R/P_i)}\\& = & e_{J}(B) \) 
Therefore we get \(F=\lbrace P_1,\ldots ,P_s\rbrace \)  and \(e_{JR_{P_i}}(R_{P_i})=e_{IR_{P_i}}(R_{P_i})\)  for all \(1\le i \le s.\)  Now since \(R\)  is quasi-unmixed and \(J\)  is an equimultiple ideal, by Theorem REF , \(J\)  is a reduction of \(I\) , which implies that \(\ell (I)=h\) . Thus by Theorem [2]}, \(\mathcal {R}_s({\bf a})\)  is Noetherian.
